A posteriori error estimates for fourth order hyperbolic control problems by mixed finite element methods

R E S E A R C H

Open Access

A posteriori error estimates for fourth order

hyperbolic control problems by mixed finite

element methods

Chunjuan Hou

_{, Zhanwei Guo}

_{and Lianhong Guo}

*_{Correspondence:} [email protected]

1_{Huashang College Guangdong} University of Finance and Economics, Guangzhou, P.R. China

Abstract

In this paper, we consider the a posteriori error estimates of the mixed finite element method for the optimal control problems governed by fourth order hyperbolic equations. The state is discretized by the orderkRaviart–Thomas mixed elements and control is discretized by piecewise polynomials of degreek. We adopt the mixed elliptic reconstruction to derive the a posteriori error estimates for both the state and the control approximations.

MSC: 49J20; 65N30

Keywords: A posteriori error estimates; Optimal control problems; Fourth order hyperbolic equations; Mixed finite element methods

1 Introduction

The finite element approximation for optimal control problems has an enormously im-portant function in the numerical approach for these problems. Scientists have studied extensively this area; see, for example, [4,12,13,21,25]. They discussed the a priori er-ror estimates using finite element approximations, such as [1,16,23], in which elliptic or parabolic problems are considered by optimal control theory. They studied adaptivity for many optimal control problems; for example, see [4,11,17,20–22].

In some optimal control problems, for the objective function containing a gradient of the state variable, we use mixed finite element methods to discretize the state equation, so that the scalar variable and its flux variable can be approximated in the same accuracy; for example, see [3]. Many scientists have addressed the mixed finite element methods for elliptic problems [6–8,14], for the first bi-harmonic equation [5], for parabolic problems [26] and for hyperbolic problems [9,15].

The purpose of this work is to discuss the a posteriori error estimates of the semidiscrete mixed finite element approximation for fourth order hyperbolic optimal control problems. Considering the fourth order hyperbolic equations by the idea of a mixed elliptic recon-struction [24], we obtain the error estimates for the state and the control approximations. The following is the model we considered:

min

u∈K∈U

1 2

T

0

y2+y–yd2+u2

dt

, (1.1)

ytt(x,t) +2y(x,t) =f(x,t) +u(x,t), x∈Ω,t∈J, (1.2)

y(x,t) = ∂y

∂n= 0, x∈∂Ω,t∈J, (1.3)

y(x, 0) =y0(x), x∈Ω, (1.4)

yt(x, 0) =y1(x), x∈Ω, (1.5)

whereΩ⊂R2_{is an open set of polygon with∂Ω.Kis inU=L}2_(J;L2_{(Ω)), a closed convex} set,J= [0,T],f,yd∈L2(J;L2(Ω)) andy0,y1∈H4(Ω).Kis defined as follows:

K=

u∈U:

T

0

Ω

u dx dt≥0

. (1.6)

Lety˜= –y,p˜= –∇yandp= –∇˜y, then (1.1)–(1.5) can be written as

min

u∈K∈U

1 2

T

0

˜y2+y–yd2+u2

dt

, (1.7)

ytt(x,t) +divp(x,t) =f(x,t) +u(x,t), x∈Ω,t∈J, (1.8)

p(x,t) = –∇˜y(x,t), x∈Ω,t∈J, (1.9)

divp˜(x,t) =y(x,˜ t), x∈Ω,t∈J, (1.10)

˜

p(x,t) = –∇y(x,t), x∈Ω,t∈J, (1.11)

y(x,t) = –p˜·n= 0, x∈∂Ω,t∈J, (1.12)

y(x, 0) =y0(x), x∈Ω, (1.13)

yt(x, 0) =y1(x), x∈Ω. (1.14)

In the paper, we adopt the standard notationWm,p_{(Ω) for Sobolev space onΩ with a}

normvm,pgiven byvpm,p:=

|α|≤mDαv p

Lp_(Ω), and a seminorm|v|m,pgiven by|v|pm,p:=

|α|=mDαv p

Lp_(Ω). We setW₀m,p(Ω) ={v∈Wm,p(Ω) :γ(Dαv)|∂Ω= 0,|α|=m}, whereγ is

the trace operator. We denoteWm,2(Ω)(W₀m,2(Ω)) byHm(Ω)(H₀m(Ω)).

We denote byLs_(0,T;Wm,p_{(Ω)) the Banach space of allL}s_{integrable functions fromJ}

intoWm,p(Ω) with normvLs_(J;Wm,p_(Ω))= (T

0 v

s

Wm,p_(Ω)dt)

1

s _fors∈_[1,∞_{), and the}

stan-dard modification fors=∞. For simplicity of presentation, we denotevLs_(J;Wm,p_(Ω))by vLs_(Wm,p). Similarly, one can define the spacesH1(J;Wm,p(Ω)) andCk(J;Wm,p(Ω)). We

can find details in [19].Cis a general positive constant independent ofh.

The rest of this paper is as follows: In Sect.2, we introduce the optimal control problems and its mixed finite element scheme. Section2ends with the definition of the mixed elliptic reconstructions, which is useful in deriving the a posteriori estimates for the fourth order hyperbolic optimal control problems in Sect.3. Finally, we make some concluding remarks in Sect.4.

2 Optimal control problems for mixed methods

A semidiscrete approximation of a mixed finite element for the optimal control problems (1.7)–(1.14) will be constructed. We set the state spacesL=L2_{(J;V), L}

Q=L2_(J;W),W=L2_{(Ω), whereV, andV}

0are defined as follows:

V=H(div;Ω) = v∈L2(Ω)2,divv∈L2(Ω),

V₀=H0(div;Ω) = v∈H(div,Ω),v·n|∂Ω= 0

.

The spaceVis a Hilbert space, its norm is defined as follows:

vH(div;Ω)=

v2_0,Ω+divv2_0,Ω1/2.

Now we introduce operators:div,∇,curlandCurl. For anyv= (v₁,v₂)∈(H1_(Ω))2 _or w∈H1(Ω),

divv=∂1v1+∂2v2, ∇w= (∂1w,∂2w), (2.1)

curlv=∂1v2–∂2v1, Curlw= (–∂2w,∂1w). (2.2)

Next, (1.7)–(1.14) can be rewritten into weak form as follows: find (p˜,y,p,y,u)∈(L0× Q×L×Q×K), such that

min

u∈K⊂U

1 2

T

0

˜y2+y–yd2+u2

dt

, (2.3)

(ytt,w) + (divp,w) = (f+u,w), ∀w∈W,t∈J, (2.4)

(p,v) – (y,˜ divv) = 0, ∀v∈V₀,t∈J, (2.5)

(divp˜,w) = (y,˜ w), ∀w∈W,t∈J, (2.6)

(p˜,v) – (y,divv) = 0, ∀v∈V,t∈J, (2.7)

y(x, 0) =y0(x), ∀x∈Ω, (2.8)

yt(x, 0) =y1(x), ∀x∈Ω. (2.9)

From [18], we know that the above optimal control problem has a unique solution

(p˜,y,p,y,˜ u), and that (p˜,y,p,y,˜ u) is the solution of (2.3)–(2.9) if and only if there is a co-state (q˜,z,q,˜z)∈(L0×Q×L×Q) such that (p˜,y,p,y,˜ q˜,z,q,z,˜ u) satisfies the following optimality conditions:

(ytt,w) + (divp,w) = (f+u,w), ∀w∈W,t∈J, (2.10)

(p,v) – (y,˜ divv) = 0, ∀v∈V,t∈J, (2.11)

(divp˜,w) = (y,˜ w), ∀w∈W,t∈J, (2.12)

(p˜,v) – (y,divv) = 0, ∀v∈V,t∈J, (2.13)

y(x, 0) =y0(x), ∀x∈Ω, (2.14)

yt(x, 0) =y1(x), ∀x∈Ω, (2.15)

(ztt,w) + (divq,w) = (y–yd,w), ∀w∈W,t∈J, (2.16)

(divq˜,w) = (y˜+z,˜ w), ∀w∈W,t∈J, (2.18)

(q˜,v) – (z,divv) = 0, ∀v∈V,t∈J, (2.19)

z(x,T) = 0, ∀x∈Ω, (2.20)

zt(x,T) = 0, ∀x∈Ω, (2.21)

T

0

(u+z,u˜–u)dt≥0, ∀˜u∈K, (2.22)

where (·,·) is the inner product ofL2_(Ω).

Kis a control constraint, so we can get a relationship betweenuandz. This relationship is important for our result.

Lemma 2.1 Let(z,u)be the solution of(2.10)–(2.22).Then we have u=max{0,zˇ}–z,where

ˇ

z= T

0

Ωz dx dt

T

0

Ω1dx dt

denotes the integral average onΩ×J of the function z.

Let Th be regular triangulations of Ω, hτ is the diameter of τ andh=maxhτ.

Fur-thermore, letEh be the set of element sides of the triangulationTh withΓh=

_E

h. Let

Vh×Wh⊂V×W denote the Raviart–Thomas space [3] associated with the

triangula-tionsThofΩ.Pkdenotes the space of polynomials of total degree no greater thank(k≥0).

LetV(τ) ={v∈P_k2(τ) +x·Pk(τ)},W(τ) =Pk(τ). We set

Vh:= vh∈V:∀τ ∈Th,vh|τ∈V(τ)

,

Wh:= wh∈W:∀τ∈Th,wh|τ∈W(τ)

,

Kh:=L2(J;Wh)∩K.

We now discretize (2.3)–(2.9). We calculate (p˜_h,yh,ph,y˜h,uh) such that

min

uh∈Kh

1 2

T

0

˜yh2+yh–yd2+uh2

dt

, (2.23)

(yhtt,wh) + (divph,wh) = (f+uh,wh), ∀wh∈Wh,t∈J, (2.24)

(p_h,v_h) – (y˜h,divvh) = 0, ∀vh∈Vh,t∈J, (2.25)

(divp˜_h,wh) = (˜yh,wh), ∀wh∈Wh,t∈J, (2.26)

(p˜_h,vh) – (yh,divvh) = 0, ∀vh∈Vh,t∈J, (2.27)

yh(x, 0) =yh0(x), ∀x∈Ω, (2.28)

yht(x, 0) =yh1(x), ∀x∈Ω, (2.29)

where yh

and (p˜_h,yh,ph,y˜h,uh) is the solution of (2.23)–(2.29) if and only if there is a co-state

(q˜_h,zh,qh,˜zh) such that the following optimality conditions hold:

(yhtt,wh) + (divph,wh) = (f+uh,wh), ∀wh∈Wh,t∈J, (2.30)

(p_h,vh) – (y˜h,divvh) = 0, ∀vh∈Vh,t∈J, (2.31)

(divp˜_h,wh) = (˜yh,wh), ∀wh∈Wh,t∈J, (2.32)

(p˜_h,vh) – (yh,divvh) = 0, ∀vh∈Vh,t∈J, (2.33)

yh(x, 0) =yh0(x), ∀x∈Ω, (2.34)

yht(x, 0) =yh1(x), ∀x∈Ω, (2.35)

(zhtt,wh) + (divqh,wh) = (yh–yd,wh), ∀wh∈Wh,t∈J, (2.36)

(q_h,v_h) – (z˜h,divvh) = 0, ∀vh∈Vh,t∈J, (2.37)

(divq˜_h,wh) = (˜yh+˜zh,wh), ∀wh∈Wh,t∈J, (2.38)

(q˜_h,vh) – (zh,divvh) = 0, ∀vh∈Vh,t∈J, (2.39)

zh(x,T) = 0, ∀x∈Ω, (2.40)

zht(x,T) = 0, ∀x∈Ω, (2.41)

T

0

(uh+zh,u˜h–uh)dt≥0, ∀˜uh∈Kh. (2.42)

For Lemma2.1, the relationship betweenuhandzhis given as follows:

uh=max{0,zˇh}–zh, (2.43)

wherezˇh=

T

0

Ωzhdx dt

T

0

Ω1dx dt

is the integral average onΩ×Jof the functionzh.

Now, we give the local definition of divh, curlh : H1(Th)2 →L2(Ω) and ∇h, Curlh :

H1_(T

h)→L2(Ω)2, such that for anyT∈Th

divhv|T:=div(v|T), curlhv|T:=curl(v|T), (2.44)

∇hv|T:=∇(v|T), Curlhv|T:=Curl(v|T). (2.45)

SetPh:W→Whto be the orthogonalL2(Ω)-projection intoWh[2], which satisfies

(Phw–w,χ) = 0, w∈W,χ∈Wh, (2.46)

Phw–w0,q≤Chtwt,q, 0≤t≤k+ 1, ifw∈W∩Wt,q(Ω), (2.47)

Phw–w–r≤Chr+twt, 0≤r,t≤k+ 1, ifw∈Ht(Ω). (2.48)

Next, introduce the Fortin projection (see [3] and [10])Πh:V→Vh, which satisfies: for

anyq∈V

div(Πhq–q),wh

= 0, ∀q∈V,wh∈Wh, (2.49)

q–Πhq0,q≤Chrqr,q, 1/q<r≤k+ 1,∀q∈V∩

Wr,q(Ω)2, (2.50)

The commuting diagram property reads

div◦Πh=Ph◦div:V→Wh and div(I–Πh)V⊥Wh, (2.52)

whereIdenotes the identity operator.

Next, the intermediate variableu˜∈Kis introduced as follows:

ytt(u),˜ w

+divp(u),˜ w= (f+u,˜ w), ∀w∈W,t∈J, (2.53)

p(u),˜ v–y(˜u),˜ divv= 0, ∀v∈V,t∈J, (2.54)

divp˜(u),˜ w=y(˜ u),˜ w, ∀w∈W,t∈J, (2.55)

˜

p(u),˜ v–y(u),˜ divv= 0, ∀v∈V,t∈J, (2.56)

y(u)(x, 0) =˜ y0(x), ∀x∈Ω, (2.57)

yt(u)(x, 0) =˜ y1(x), ∀x∈Ω, (2.58)

ztt(u),˜ w

+divq(u),˜ w=y(u) –˜ yd,w

, ∀w∈W,t∈J, (2.59)

q(u),˜ v–z(˜u),˜ divv= 0, ∀v∈V,t∈J, (2.60)

divq˜(u),˜ w=y(˜ u) +˜ z(˜u),˜ w, ∀w∈W,t∈J, (2.61)

˜

q(u),˜ v–z(u),˜ divv= 0, ∀v∈V,t∈J, (2.62)

z(u)(x,˜ T) = 0, ∀x∈Ω, (2.63)

zt(u)(x,˜ T) = 0, ∀x∈Ω. (2.64)

Next, we present mixed elliptic constructions (p˜¯,y,¯ p¯,y,˜¯ q˜¯,z,¯ q¯,z)˜¯ ∈(V×W)4_:

(p˜¯,v) – (y,¯ divv) = 0, ∀v∈V, (2.65)

(divp˜¯,w) = (y˜h,w), ∀w∈W, (2.66)

(p¯,v) – (y,˜¯ divv) = 0, ∀v∈V, (2.67)

(divp¯,w) = (f +uh–yhtt,w), ∀w∈W, (2.68)

(q˜¯,v) – (¯z,divv) = 0, ∀v∈V, (2.69)

(divq˜¯,w) = (z˜h+y˜h,w), ∀w∈W, (2.70)

(q¯,v) – (˜¯z,divv) = 0, ∀v∈V, (2.71)

(divq¯,w) = (yh–yd–zhtt,w), ∀w∈W. (2.72)

For simplicity of presentation, we resolve the errors in following forms:

˜

p–p˜_h=p˜–p˜(uh) +p˜(uh) –p˜¯+p˜¯–p˜h=r1+e1+η1, (2.73)

y–yh=y–y(uh) +y(uh) –y¯+¯y–yh=r2+e2+η2, (2.74)

p–p_h=p–p(uh) +p(uh) –p¯+p¯–ph=r3+e3+η3, (2.75)

˜

˜

q–q˜_h=q˜–q˜(uh) +q˜(uh) –q˜¯+q˜¯–q˜h=r5+e5+η5, (2.77)

z–zh=z–z(uh) +z(uh) –z¯+¯z–zh=r6+e6+η6, (2.78)

q–q_h=q–q(uh) +q(uh) –q¯+q¯–qh=r7+e7+η7, (2.79)

˜

z–z˜h=z˜–z(u˜ h) +z(u˜ h) –z˜¯+˜¯z–˜zh=r8+e8+η8. (2.80)

From mixed elliptic reconstructions [24], we derive the error estimates as below.

Lemma 2.2 ([8, 14]) For Raviart–Thomas elements, there exists a positive constant C which depends the domainΩ,the shape regularity of the elements and polynomial degree k such that

η2 ≤C

h1+min{1,k}(divp˜_h–y˜h)+ min wh∈Wh

h(p˜_h–∇hwh), (2.81)

η2t ≤C

h1+min{1,k}(divp˜_ht–y˜ht)+ min wh∈Wh

h(p˜_ht–∇hwh), (2.82)

η2tt ≤C

h1+min{1,k}(divp˜_htt–y˜htt)+ min wh∈Wh

h(p˜_htt–∇hwh), (2.83)

η2ttt ≤C

h1+min{1,k}_(divp˜

httt–y˜httt)+ min wh∈Wh

h(p˜_httt–∇hwh), (2.84)

η4 ≤C

h1+min{1,k}(yhtt+divph–f –uh)+ min wh∈Wh

h(p_h–∇hwh), (2.85)

η4t ≤C

h1+min{1,k}(yhtt+divph–f–uh)t+ min wh∈Wh

h(p_ht–∇hwh), (2.86)

η4tt ≤C

h1+min{1,k}(yhtt+divph–f –uh)tt+ min wh∈Wh

h(p_htt–∇hwh), (2.87)

η6 ≤C

h1+min{1,k}(divq˜_h–y˜h–z˜h)+ min wh∈Wh

_h(q_˜h–∇hwh), (2.88)

η6t ≤C

h1+min{1,k}(divq˜_ht–y˜ht–z˜ht)+ min wh∈Wh

h(q˜_ht–∇hwh), (2.89)

η6tt ≤C

h1+min{1,k}(divq˜_htt–y˜htt–z˜htt)+ min wh∈Wh

h(q˜_htt–∇hwh), (2.90)

η8 ≤C

h1+min{1,k}(zhtt+divqh–yh+yd)+ min wh∈Wh

h(q_h–∇hwh), (2.91)

η8t ≤C

h1+min{1,k}(zhtt+divqh–yh+yd)t+ min wh∈Wh

h(q_ht–∇hwh), (2.92)

η1 ≤Ch 1 2_J(p˜

h·t)_0,Γh+hcurlhp˜h+h(divp˜h–y˜h), (2.93)

η3 ≤Ch 1 2_J(p

h·t)_0,Γh+hcurlhph+h(yhtt+divph–f –uh), (2.94)

η5 ≤Ch 1 2_J(q˜

h·t)0,Γh+hcurlhq˜h+h(divq˜h–y˜h–y˜h), (2.95)

η7 ≤Ch 1 2_J(q

h·t)0,Γh+hcurlhqh+

_h(zhtt+divq_h–yh+yd), (2.96)

divη1 ≤Cdivp˜h–y˜h,divη3 ≤Cyhtt+divph–f –uh, (2.97)

divη5 ≤Cdivq˜h–y˜h–z˜h,divη7 ≤Czhtt+divqh–yh+yd, (2.98)

where∇handcurlhhave been defined in(2.44)–(2.45),J(v·t)denotes the jump ofv·tacross

3 Error estimation of optimal control

In this part, a posteriori error estimation of optimal control problems shall be given. From (2.53)–(2.56) and (2.65)–(2.68), we obtain the error equations

(e1,v) – (e2,divv) = 0, ∀v∈V, (3.1)

(dive1,w) = (e4+η4,w), ∀w∈W, (3.2)

(e3,v) – (e4,divv) = 0, ∀v∈V, (3.3)

(e2tt,w) + (dive3,w) = –(η2tt,w), ∀w∈W. (3.4)

Lemma 3.1 Let e1–e4satisfy(3.1)–(3.4).Then we have

e2L∞(J;L2_(Ω))+e1_L∞_(J;H(div;Ω))+e4L∞(J;L2_(Ω))+e3_L∞_(J;H(div;Ω))

≤Cη4tL2_(J;L2_(Ω))+y₁–yh₁+η_2t(0)+y₀+y˜_h(0)+η_4L∞_(J;L2_(Ω))

+η2ttL∞(J;L2_(Ω))+η_2tttL2_(J;L2_(Ω))+η_4ttL2_(J;L2_(Ω))+y0–yh₀

+divη3(0)+2y0–divph(0)

+y1+y˜ht(0)+η4t(0)) +η2(0)). (3.5)

Proof Differentiating (3.1)–(3.2) with respect tot, we get

(e1t,v) – (e2t,divv) = 0, ∀v∈V, (3.6)

(dive1t,w) = (e4t+η4t,w), ∀w∈W. (3.7)

We choosev=e3in (3.6),w= –e4in (3.7),v= –e1tin (3.3) andw=e2tin (3.4), separately,

then add up the four equations and obtain

(e2t,e2tt) + (e4t,e4) = –(η4t,e4) – (η2tt,e2t). (3.8)

We integrate (3.8) from 0 tot, use Gronwall’s inequality and the Cauchy inequality, then we obtain

e4L∞(J;L2_(Ω))+e_2tL∞_(J;L2_(Ω))

≤Cη4tL2_(J;L2_(Ω))+η_2ttL2_(J;L2_(Ω))+e2_t(0)+e4(0), (3.9)

where

e2t(0)≤y1–yh1+η2t(0), (3.10)

e4(0)≤y0+y˜h(0)+η4(0). (3.11)

Note that t

0

then we have

e2 ≤Ce2tL∞(J;L2_(Ω))+e2(0)

≤Ce2tL∞(J;L2_(Ω))+y0–yh₀+η2(0). (3.12)

Lettingv=e1in (3.1),w=e2in (3.2),v=e3in (3.3) andw=e4in (3.4), respectively. We get

e1L∞(J;L2_(Ω))≤ η_4L∞_(J;L2_(Ω))+e4_L∞_(J;L2_(Ω))+e2_L∞_(J;L2_(Ω)), (3.13)

e3L∞(J;L2_(Ω))≤ e2_ttL∞_(J;L2_(Ω))+η_2ttL∞_(J;L2_(Ω))+e4_L∞_(J;L2_(Ω)). (3.14)

Differentiating (3.3)–(3.4) and (3.6)–(3.7) with respect tot, we get

(e3t,v) – (e4t,divv) = 0, ∀v∈V, (3.15)

(e2ttt,w) + (dive3t,w) = –(η2ttt,w), ∀w∈W, (3.16)

(e1tt,v) – (e2tt,divv) = 0, ∀v∈V, (3.17)

(dive1tt,w) = (e4tt+η4tt,w), ∀w∈W. (3.18)

We choosev= –e1tt in (3.15),w=e2ttin (3.16),v=e3tin (3.17),w= –e4t in (3.18),

sepa-rately. We derive the following after addition for four equations:

(e2ttt,e2tt) + (e4tt,e4t) = –(η4tt,e4t) – (η2ttt,e2tt). (3.19)

Similar to (3.9), we derive

e2ttL∞(J;L2_(Ω))+e4_tL∞_(J;L2_(Ω))

≤Cη2tttL2_(J;L2_(Ω))+η_4ttL2_(J;L2_(Ω))+e2_tt(0)+e4_t(0). (3.20)

Takingt= 0 andw=e2tt(0) in (3.4) leads to

e2tt(0)≤dive3(0)+η2tt(0)

≤divη3(0)+η2tt(0)+2y0–divph(0). (3.21)

Note that

e4t(0)≤y˜t(uh)(0) –y˜ht(0)+η4t(0)

≤y1+y˜ht(0)+η4t(0). (3.22)

At last, settingw=dive1andw=dive3in (3.2) and (3.4), we get

dive1L∞(J;L2_(Ω))≤ η_4L∞_(J;L2_(Ω))+e_4L∞_(J;L2_(Ω)), (3.23)

dive3L∞(J;L∞(Ω))≤ e2ttL∞(J;L2_(Ω))+η_2ttL∞_(J;L2_(Ω)). (3.24)

By (2.59)–(2.62) and (2.69)–(2.72), we obtain the error equations

(e5,v) – (e6,divv) = 0, ∀v∈V, (3.25)

(dive5,w) = (e4+e8+η4+η8,w), ∀w∈W, (3.26)

(e7,v) – (e8,divv) = 0, ∀v∈V, (3.27)

(e6tt,w) + (dive7,w) = (η6tt+η2+e2,w), ∀w∈W. (3.28)

Lemma 3.2 Let e5–e8satisfy(3.25)–(3.28).Then we get

e6L∞(J;L2_(Ω))+e_5L∞_(J;H(div;Ω))+e8L∞(J;L2_(Ω))+e_7L2_{(J;H(div;Ω))}

≤Cη4L∞(J;L2_(Ω))+η_4tL2_(J;L2_(Ω))+η_4ttL2_(J;L2_(Ω))+η_2L2_(J;L2_(Ω))

+η6ttL2_(J;L2_(Ω))+η_8L∞_(J;L2_(Ω))+η_8tL2_(J;L2_(Ω))+η_2tttL2_(J;L2_(Ω))

+e4L∞(J;L2_(Ω))+e_2L2_(J;L2_(Ω))+y₁+y˜_ht(0)+η_4t(0)

+divη3(0)+η2tt(0)+2y0–divph(0). (3.29)

Proof At first, settingt=T in (2.69)–(2.70) and (3.25)–(3.26), we derive

e6(T) =e6t(T) = 0 (3.30)

and

e8(T)≤Ce4(T)+η4(T)+η8(T). (3.31)

Differentiating (3.25)–(3.26) with respect tot, we obtain

(e5t,v) – (e6t,divv) = 0, ∀v∈V, (3.32)

(dive5t,w) = (e4t+e8t+η4t+η8t,w), ∀w∈W. (3.33)

Letv= –e7in (3.32),w=e8in (3.33),v=e5t in (3.27) andw= –e6tin (3.28), separately.

After adding up the new equations, we have

–(e6tt,e6t) – (e8t,e8) = (e4t+η4t+η8t,e8) – (η6tt+η2+e2,e6t). (3.34)

Integrating (3.34) fromttoT, from (3.30), Gronwall’s inequality and the Cauchy inequal-ity, it is easy to see that

e6tL∞(J;L2_(Ω))+e8_L∞_(J;L2_(Ω))

≤Cη4tL2_(J;L2_(Ω))+e4_tL2_(J;L2_(Ω))+η_8tL2_(J;L2_(Ω))

+e2L2_(J;L2_(Ω))+η_2L2_(J;L2_(Ω))+η_6ttL2_(J;L2_(Ω))+e8(T). (3.35)

Lettingv=e7in (3.27), we get

Next, for (3.25), we differentiate two times with respect tot, and setv=e7. for (3.26), we also differentiate two times with respect tot, and setw=e8. For (3.27), we setv=e5tt.

For (3.28), we setw=dive7. Combining the new four equalities, we derive

dive7L2_(J;L2_(Ω))≤C

e2L2_(J;L2_(Ω))+η_6ttL2_(J;L2_(Ω))+η_2L2_(J;L2_(Ω))+η_4tL∞_(J;L2_(Ω))

+η8tL∞(J;L2_(Ω))

+e4L2_(J;L2_(Ω))) +e8_tL2_(J;L2_(Ω))). (3.37)

At last, similar to (3.13)–(3.14), (3.23)–(3.24) and (3.36), we have

e5L∞(J;H(div;Ω))≤C

e4L∞(J;L2_(Ω))+e_8L∞_(J;L2_(Ω))

+η4L∞(J;L2_(Ω))+η_8L∞_(J;L2_(Ω))

+e6L∞(J;L2_(Ω))), (3.38)

e7L2_{(J;H(div;Ω))}

≤Ce2L2_(J;L2_(Ω))+η_6ttL2_(J;L2_(Ω))+η_2L2_(J;L2_(Ω))+η_4tL∞_(J;L2_(Ω))

+η8tL∞(J;L2_(Ω))+e4_L2_(J;L2_(Ω))

+e8tL2_(J;L2_(Ω))+e8_L2_(J;L2_(Ω))). (3.39)

Thus, combining Lemma3.1, (3.31), (3.35)–(3.40) and (3.5), we complete the proof.

Choosingu˜=uandu˜=uhin (2.53)–(2.64), respectively, we have the error equations

(r1,v) – (r2,divv) = 0, ∀v∈V, (3.40)

(divr1,w) = (r4,w), ∀w∈W, (3.41)

(r3,v) – (r4,divv) = 0, ∀v∈V, (3.42)

(r2tt,w) + (divr3,w) = (u–uh,w), ∀w∈W, (3.43)

(r5,v) – (r6,divv) = 0, ∀v∈V, (3.44)

(divr5,w) = (r4+r8,w), ∀w∈W, (3.45)

(r7,v) – (r8,divv) = 0, ∀v∈V, (3.46)

(r6tt,w) + (divr7,w) = (r2,w), ∀w∈W. (3.47)

Similar to Lemmas3.1and3.2, Lemma3.3is given below.

Lemma 3.3 Let r1–r8satisfy(3.40)–(3.47).Then we have

r2tL∞(J;L2_(Ω))+r4_L∞_(J;L2_(Ω))+r1_L∞_(J;H(div;Ω))+r3L2_{(J;H(div;Ω))}

≤Cu–uhL2_(J;L2_(Ω)), (3.48)

r4tL∞(J;L2_Ω))+r_6tL∞_(J;L2_Ω))+r_8L∞_(J;L2_(Ω))

≤u(0) –uh(0)

+Cu–uhL2_(J;L2_(Ω))+(u–u_h)_t

r5L∞(J;H(div;Ω))+r7L2_{(J;H(div;Ω))}

≤u(0) –uh(0)

+Cu–uhL2_(J;L2_(Ω))+(u–uh)t_L2_(J;L2_(Ω)), (3.50)

whereis an arbitrary small positive constant.

Lemma 3.4 [15]Let(p˜,y,p,y,˜ q˜,z,q,z,˜ u)and(p˜_h,yh,ph,y˜h,q˜h,zh,qh,uh)be the solutions

of(2.10)–(2.22)and(2.30)–(2.42),respectively.Suppose that(uh+zh)|τ ∈H1(τ)and that

there exists w∈Khsuch that

u–uhL2_(J;L2_(Ω))≤C(θ+z_h–z(u_h)

L2_(J;L2_(Ω)), (3.51)

where

θ=

T

0

τ

h2_τ|uh+zh|2_H1_(τ)dt 1

2 .

Now, by Lemmas3.1–3.3, the important result of this paper is given as follows.

Theorem 3.1 Let(y,p˜,˜y,p,z,q˜,z,˜ q,u)and(yh,p˜h,˜yh,ph,zh,q˜h,z˜h,qh,uh)be the solutions

of(2.9)–(2.19)and(2.26)–(2.36),respectively.Then we have

u–uhL∞(J;L2_(Ω))+y–y_hL∞_(J;L2_(Ω))+˜y–y˜_hL∞_(J;L2_(Ω))

+˜p–p˜_hL∞_(J;H(div;Ω))+p–phL∞(J;H(div;Ω))+z–zhL∞(J;L2_(Ω))

+˜z–z˜hL∞(J;L2_(Ω))+˜q–q˜_hL∞_(J;H(div;Ω))+q–qhL2_{(J;H(div;Ω))}

≤Cη1L∞(J;H(div;Ω))+η3L∞(J;H(div;Ω))+η5L∞(J;H(div;Ω))+η7L2_{(J;H(div;Ω))}

+η4L∞(J;L2_(Ω))+η_4tL2_(J;L2_(Ω))+η_4ttL2_(J;L2_(Ω))+η_2L2_(J;L2_(Ω))

+η2ttL∞(J;L2_(Ω))+η_2tttL2_(J;L2_(Ω))+η_6L∞_(J;L2_(Ω))+η_6tL2_(J;L2_(Ω))

+η6ttL2_(J;L2_(Ω))+η_8L∞_(J;L2_(Ω))+η_8tL2_(J;L2_(Ω))+y0–yh₀

+y0+y˜h(0)+y1–yh1+divη3(0)+2y0–divph(0)

+y1+y˜ht(0)+η2(0)+η2t(0)+η4t(0). (3.52)

Proof From Lemma2.1and (2.43), we have

u(0) –uh(0)≤ u–uhL∞(J;L2_(Ω))≤Cz–z_hL∞_(J;L2_(Ω)), (3.53) (u–uh)t_L2_(J;L2_(Ω))=(z–zh)t_L2_(J;L2_(Ω))

≤ e6tL2_(J;L2_(Ω))+η_6tL2_(J;L2_(Ω))+r6_tL2_(J;L2_(Ω)). (3.54)

For sufficiently small, using Lemmas3.1–3.3and (3.35), (3.53)–(3.54), we complete the

4 Conclusion and future work

In the article, using semidiscrete Raviart–Thomas mixed finite element methods, we stud-ied fourth order hyperbolic equations of quadratic problems for optimal control, and then got the posteriori error estimates. In subsequent work, an a posteriori estimation will be considered by a fully discrete approximation of the mixed finite element. Of course, the error estimates of the same problems certainly also can be discussed with nonlinear fourth order hyperbolic equations.

Acknowledgements

The authors express their thanks to the referees for their helpful suggestions, which led to improvements of the presentation.

Funding

This work is supported by Youth Innovative Talents Project (Natural Science) of research on humanities and social sciences in Guangdong normal university (2017KQNCX265), The issue for the 13th Five-Year plan for the development of philosophy and social sciences in Guangzhou of 2018 (2018GZGJ168), School projects of Huashang College Guangdong University of Finance and Economics.

Availability of data and materials

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

CH, ZG and LG participated in the sequence alignment and drafted the manuscript. All authors read and approved the final manuscript.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 20 July 2017 Accepted: 7 May 2019

References

1. Arada, N., Casas, E., Tröltzsch, F.: Error estimates for the numerical approximation of a semilinear elliptic control problem. Comput. Optim. Appl.23, 201–229 (2002)

2. Babuška, I., Strouboulis, T.: The Finite Element Method and Its Reliability. Oxford University Press, Oxford (2001) 3. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, Berlin (1991)

4. Brunner, H., Yan, N.: Finite element methods for optimal control problems governed by integral equations and integro-differential equations. Numer. Math.101, 1–27 (2005)

5. Cao, W., Yang, D.: Ciarlet–Raviart mixed finite element approximation for an optimal control problem governed by the first bi-harmonic equation. J. Comput. Appl. Math.233(2), 372–388 (2009)

6. Chen, Y.: Superconvergence of quadratic optimal control problems by triangular mixed finite elements. Int. J. Numer. Methods Eng.75, 881–898 (2008)

7. Chen, Y., Huang, Y., Liu, W.B., Yan, N.N.: Error estimates and superconvergence of mixed finite element methods for convex optimal control problems. J. Sci. Comput.42, 382–403 (2009)

8. Chen, Y., Liu, W.B.: A posteriori error estimates for mixed finite element solutions of convex optimal control problems. J. Comput. Appl. Math.211, 76–89 (2008)

9. Chen, Y., Sun, C.M.: Error estimates and superconvergence of mixed finite element methods for fourth order hyperbolic control problems. Appl. Math. Comput.244, 642–653 (2014)

10. Douglas, J., Roberts, J.E.: Global estimates for mixed finite element methods for second order elliptic equations. Math. Comput.44, 39–52 (1985)

11. Gong, W., Yan, N.: A posteriori error estimate for boundary control problems governed by the parabolic partial differential equations. J. Comput. Math.27, 68–88 (2009)

12. Haslinger, J., Neittaanmaki, P.: Finite Element Approximation for Optimal Shape Design. Wiley, Chichester (1989) 13. Hou, L., Turner, J.C.: Analysis and finite element approximation of an optimal control problem in electrochemistry

with current density controls. Numer. Math.71, 289–315 (1995)

14. Hou, T.: Error estimates of mixed finite element approximations for a class of fourth order elliptic control problems. Bull. Korean Math. Soc.4(50), 1127–1144 (2013)

15. Hou, T.: A posterioriL∞(L2_{)-error estimates of semidiscrete mixed finite element methods for hyperbolic optimal} control problems. Bull. Korean Math. Soc.50, 321–341 (2013)

16. Knowles, G.: Finite element approximation of parabolic time optimal control problems. SIAM J. Control Optim.20, 414–427 (1982)

17. Li, R., Liu, W., Ma, H., Tang, T.: Adaptive finite element approximation of elliptic control problems. SIAM J. Control Optim.41, 1321–1349 (2002)

19. Lions, J., Magenes, E.: Non Homogeneous Boundary Value Problems and Applications. Grandlehre, vol. 181. Springer, Berlin (1972)

20. Liu, W., Ma, H., Tang, T., Yan, N.: A posteriori error estimates for discontinuous Galerkin time-stepping method for optimal control problems governed by parabolic equations. SIAM J. Numer. Anal.42, 1032–1061 (2004) 21. Liu, W., Yan, N.: A posteriori error estimates for convex boundary control problems. SIAM J. Numer. Anal.39, 73–99

(2001)

22. Liu, W., Yan, N.: A posteriori error estimates for optimal control problems governed by Stokes equations. SIAM J. Numer. Anal.40, 1850–1869 (2003)

23. Mcknight, R., Bosarge, Jr., W.: The Ritz–Galerkin procedure for parabolic control problems. SIAM J. Control Optim.11, 510–524 (1973)

24. Memon, S., Nataraj, N., Pani, A.K.: An a posteriori error analysis of mixed finite element Galerkin approximations to second order linear parabolic problems. SIAM J. Numer. Anal.50, 1367–1393 (2012)

25. Neittaanmaki, P., Tiba, D.: Optimal Control of Nonlinear Parabolic Systems: Theory, Algorithms and Applications. Dekker, New York (1994)